Raymond O. Collier

A Framework for the Development of Measurement Instruments for Evaluating the Introductory Statistics Course

Abstract This paper proposes a framework for the development of instruments to measure content learning and problem-solving skills for the introductory statistics course. This framework is based upon a model of the problem-solving process central to statistical reasoning. The framework defines and interrelates six measurement tasks: (1) subjective reports; (2) reports concerning truth, falsity, or equivalence; (3) supply the appropriate missing information in a message; (4) answer a question based upon a specific message; (S) reproduce a message; and (6) carry out a procedure.

Analysis of variance for correlated observations

Two different linear models are presented for the four-dimensional classification system in which correlations exist between certain pairs of observations. Except for the assumption of correlated observations, classical assumptions associated with classification systems are made. The models considered are modifications of those which underlie the split-plot design and the split-split-plot design. In the first model the correlations between observations of the levels of one dimension are all set equal toρ. In the second model the observations of the levels of one dimension are assumed correlated to degreeρ1, whereas the observations of a second dimension are correlated to degreeρ2. Analyses for the two models and tests of hypotheses for various parameters are indicated.Two different linear models are presented for the four-dimensional classification system in which correlations exist between certain pairs of observations. Except for the assumption of correlated observations, classical assumptions associated with classification systems are made. The models considered are modifications of those which underlie the split-plot design and the split-split-plot design. In the first model the correlations between observations of the levels of one dimension are all set equal toρ. In the second model the observations of the levels of one dimension are assumed correlated to degreeρ1, whereas the observations of a second dimension are correlated to degreeρ2. Analyses for the two models and tests of hypotheses for various parameters are indicated.

An Empirical Study into Factors Affecting the F-Test Under Permutation for the Randomized Block Design

Existing theoretical and empirical evidence indicates that, except for small samples, the null and non-null distributions of the variance ratios involved in the analysis of the randomized block design, based upon normal theory agree reasonably well with those under permutation theory. The present paper investigated the combined effect of block-treatment interaction and block-variance heterogeneity upon this agreement. Using a normal theory α cut-off point, the empirical probability of type I error under a permutation basis was markedly reduced by this combination of factors, and empirical power was also reduced. Additionally, the combined effect of kurtosis and block-variance heterogeneity was studied. The results indicated that kurtosis of the basic data had negligible effect upon either the null or non-null distribution of the variance ratios involved.

Tests of hypothesis in a repeated measures design from a permutation viewpoint

Repeated measures designs have been widely employed in psychological experimentation, however, such designs have rarely been analyzed by means of permutation procedures. In the present paper certain aspects of hypothesis tests ina particular repeated measures design (one non-repeated factor (A) and one repeated factor (B) withK subjects per level ofA) were investigated by means of permutation rather than sampling processes. The empirical size and power of certain normal theoryF-tests obtained under permutation were compared to their nominal normal theory values. Data sets were established in which various combinations of kurtosis of subject means and intra-subject variance heterogeneity existed in order that their effect upon the agreement of these two models could be ascertained. The results indicated that except in cases of high intra-subject variance heterogeneity, the usualF-tests onB andAB exhibited approximately the same size and power characteristics whether based upon a permutation or normal theory sampling basis.

The Effects of Subject-Treatment Interaction in a Randomized Block, One Factor Experiment

IN experiments employing individual subjects as experimental units, a useful and popular design is the generalized randomized block design, so called by Wilk and Kempthorne (1956) and Wilk (1953, 1955). In the behavioral sciences this design is sometimes referred to as the treatments-by-levels design, e.g., as employed by Lindquist (1953). Briefly stated the design consists of randomly administering treatments (in this paper belonging to a single factor) to subjects contained within each of several blocks, i.e., the several levels or intervals of a nonexperimental variable, which may be either quantitative or not. The optimal nature of such a design has been discussed by Wilk (1955), Lindquist (1953), and Feldt (1958), among others. The general desire is an increase in precision in the estimation of treat-

Some Applications of the Method of Pivotal Condensation in Statistical Analysis

THE EXISTENCE of a particular method of statistical analysis is only one of the necessary conditions for its utilization in any applied field. Equally important is the ease of calculation of the quantities involved in the method. The major purpose of this paper is that of considering the latter problem in connection with the Generalized Distance Statistic and related quantities. Pivotal condensation, the calculation technique to be illus trated, was introduced by Rao (3,4, 5). It is in tended, therefore, that this paper in part be a clarification of Raos original work.

A note on methods of indexing associative relatedness

A method of indexing response relations among stimulus words on a word association test is presented utilizing the concept of mathematical sets. Two coefficients of associative relatedness are proposed and some relationships between these and a number of other coefficients are given.

A Note On The Multiple Regression Technique For Deleting Variables In The Discriminant Function

IN AN EARLY PAPER on taxonomic problems, Fisher, [l, 184] showed that the linear discriminant function for two groups can be conveniently ob tained by constructing a certain multiple regression function. Fishers formulation employed the dis criminant variables as the independent regression variables and introduced a dichotomous dummy var iable as the dependent variable identifying group membership. Since its introduction in the litera ture, use of the regression function as a device for computing the discriminant function has been made by many researchers and discussed by several wri ters (see [4]) The preference of the regression ap proach among psychologists and educators seems to be due to the existence of numerous computer pro grams for rapid computation and also to their fa miliarity with regression methods. The purpose of this paper is to consider a method presented in multiple regression for variable selec tion or deletion for the two group discrimination problem. The actual circumstances under which the method may be used in a strict sense seem to be relatively unknown to researchers in educational fields.